Triangulated characterizations of singularities

TL;DR

This paper introduces triangulated invariants to characterize various singularities in algebraic geometry, providing explicit computations and bounds for derived categories related to these singularities.

Contribution

It develops a new invariant called 'level' within triangulated categories to measure and analyze singularity types, with explicit calculations for specific schemes and applications to Rouquier dimension.

Findings

Computed the 'level' invariant for reduced Nagata schemes of dimension one.

Derived upper bounds for Rouquier dimension in specific cases.

Linked singularity properties to invariants in derived categories.

Abstract

This work presents a range of triangulated characterizations for important classes of singularities such as derived splinters, rational singularities, and Du Bois singularities. An invariant called 'level' in a triangulated category can be used to measure the failure of a variety to have a prescribed singularity type. We provide explicit computations of this invariant for reduced Nagata schemes of Krull dimension one and for affine cones over smooth projective hypersurfaces. Furthermore, these computations are utilized to produce upper bounds for Rouquier dimension on the respective bounded derived categories.